Introduction to Data ScienceGIRI NARASIMHAN, SCIS, FIU

Time Series

CAP 5510 / CGS 5166

Time Series Analysis Applications

! Economic Forecasting ! Sales Forecasting ! Budgetary Analysis ! Stock Market Analysis ! Yield Projections

! Process and Quality Control ! Inventory Studies ! Workload Projections ! Utility Studies ! Census Analysis

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https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4.htm

CAP 5510 / CGS 5166

Average as an EstimateSuppl $ Error ES

1 9 -1 12 8 -2 43 9 -1 14 12 2 45 9 -1 16 12 2 47 11 1 18 7 -3 99 13 3 910 9 -1 111 11 1 112 10 0 0

Avg Est

7 9 10 12

SSE 144 48 36 84MSE 12 4 3 7

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" Is Average a good estimate? " Yes, it minimizes Mean

Square Error (MSE)

CAP 5510 / CGS 5166

Is Average a good predictor?

! Simple average is a bad predictor ❑ Example on right: average

does not show trend nor predict future

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CAP 5510 / CGS 5166

Moving Average vs Average

! Window size = 3 ! MSE = 3.0 ! Moving MSE = 2.42 ! Moving Average (MA) has

lower error than Average

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Suppl $ MA Error ES1 9      2 8      

3 9 8.667 0.333 0.111

4 12 9.667 2.333 5.444

5 9 10.000 -1.000 1.000

6 12 11.000 1.000 1.000

7 11 10.667 0.333 0.111

8 7 10.000 -3.000 9.000

9 13 10.333 2.667 7.111

10 9 9.667 -0.667 0.444

11 11 11.000 0 0

12 10 10.000 0 0

CAP 5510 / CGS 5166

Double Moving Averages

! Compute moving average of moving average using same window size

! Errors can be reduced further

! Do a linear regression with single and double MA to forecast

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CAP 5510 / CGS 5166

Exponential Smoothing

! MA gives equal weight to all items in window

! Exp Smoothing assigns exponentially decreasing weights for older items

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CAP 5510 / CGS 5166

Smoothing/Averaging/Filtering

! Need to remove natural variations in data ! Shows trends unhindered by local variations

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CAP 5510 / CGS 5166

Smoothing and Forecasting

! Exponential Smoothing ❑ (y1, y2, …, yn) = sequence of observations

❑ (S1, S2, …, Sn) = smoothed observations

❑ St = ⍺ yt-1 + (1- ⍺)St-1

! Forecasting ❑ St+1 = St + ⍺⍷t

❑ where ⍷t is the forecast error

! More complex forecasting

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CAP 5510 / CGS 5166

Analysis

! Check for stationarity ! Check for Trends (seasonality) ! Check for non-constant variance

❑ Trends in transformed data ▪ Log transform

! Check for Randomness ❑ Autocorrelation plots

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CAP 5510 / CGS 5166

Autocorrelation: Box-Jenkins

! Attempt to find a regression connecting Xt with one or more prior values ❑ Write down Xt as a linear combination of Xt-1, Xt-2, …, Xt-p with additive “white noise”

and mean

! R-code available

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Giri Narasimhan

Autocorrelations

! Are the data random? ! Is an observation related to an adjacent observation? observation

twice-removed? (etc.) ! Is the observed time series white noise? ! Is the observed time series sinusoidal? ! Is the observed time series autoregressive? ! What is an appropriate model for the observed time series? ! Is the model: Y = constant + error valid and sufficient? [Random data]

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Giri Narasimhan

Sinusoidal curves

! The data come from an underlying sinusoidal model.

! Note alternating sequence of positive and negative spikes.

! Spikes are not decaying.

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Giri Narasimhan 6/26/18

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CAP 5510 / CGS 5166

Autocorrelation Plots

! Plot shows mixture of ❑ exponentially decaying ❑ damped sinusoidals

! Need autoregressive model with order > 1

! Partial autocorrelation plot needed

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Giri Narasimhan

Interpreting patterns

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Pattern What the pattern indicates Example

Large spike at lag 1 that decreases after a few lags.

An autoregressive term in the data. Use the partial autocorrelation function to determine the order of the autoregressive term.

Large spike at lag 1 followed by a decreasing wave that alternates between positive and negative correlations.

A higher order autoregressive term in the data. Use the partial autocorrelation function to determine the order of the autoregressive term.

Significant correlations at the first or second lag, followed by correlations that are not significant.

A moving average term in the data. The number of significant correlations indicates the order of the moving average term.

CAP 5510 / CGS 5166

Spectral Plots

! Useful to analyze plots with complex cyclical structures

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OutliersFROM JOHNSON & WICHERN, APPLIED MULTIVARIATE STATISTICAL ANALYSIS, 6TH ED

CAP 5510 / CGS 5166

Canadian Hockey

! Kids trained early; Leagues for age groups ! Most talented get on Major Jr A league team& compete for

Memorial Cup ! What makes a top-notch hockey player? ! Soccer, Baseball, Cricket, Swimming, Gymnastics

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Jan 8 Feb 3 Mar 3 Apr 3

May 2 Jun 1 Jul Aug 2

Sep 1 Oct 1 Nov Dec 1

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CAP 5510 / CGS 5166

Canadian Hockey Players

! Cutoff birthdate is the key ! Only accept kids who are not yet 10 on Jan 1 ! January Kids matured almost one extra year over December kids

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CAP 5510 / CGS 5166

Detecting Outliers

! Visual detection

! Harder in multivariate case. Why? ❑ May be univariate or multivariate outlier

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CAP 5510 / CGS 5166

Bivariate Outliers

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CAP 5510 / CGS 5166

Multivariate Outliers

! Some outliers are hard to detect ! Look for large values of

❑

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CAP 5510 / CGS 5166

! p variables ! n items/samples

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CAP 5510 / CGS 5166

Sample Covariance & Correlation

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Basic Descriptive Statistics

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CAP 5510 / CGS 5166

Outlier detection

! Dot plots for each variable ! Scatter plot for each pair of variables ! Calculate z-values and examine for outliers

! Calculate gen sq distances & look for outliers

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CAP 5510 / CGS 5166

Spotting outliers from z-values

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CAP 5510 / CGS 5166

Spotting outliers from Gen. Distance values

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CAP 5510 / CGS 5166

Harder to spot them on scatter plots!

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CAP 5510 / CGS 5166

Other Transforms for Normality

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